EPUB is a common ebook format around the web, but the Kindle can't read it natively. You can convert .epub files to Mobi files for the Kindle to read.
The key is a free piece of software called Calibre. Calibre can reformat almost any kind of ebook for almost any ebook reader, although cracking the copy protection on books purchased from Amazon or certain other stores is forbidden.
Calibre is available for Windows and Macs. When you set up the software, it asks you for your model of ebook reader, and a pop-up suggests sending the books by email. Fill in the email address for your Kindle and the authorized email address that you've set to send documents to your Kindle with.
Once you set up Calibre, click on Add Books and pick out any free ebook files you've downloaded. They'll appear in the Calibre library. Now click on an entry in your Calibre library and choose Convert Books from the top bar. In the upper right-hand corner, make sure to set the Output Format to Mobi. Click OK, and wait until the Jobs number at the lower-right of the main screen goes from 1 down to 0.
Now select the title again and click Save to Disk. Pick the folder you want to save it in. It'll create a subfolder with the name of the author. In Explorer or Finder, drill into that folder until you find the juicy Mobi file within, which you can email to your Kindle or drag and drop onto it using a USB cable.